VISUALIZING EXTERIOR CALCULUS GREGORY BIXLER Abstract. Exterior calculus, broadly, is the structure of differential forms. These are usually presented algebraically, or as computational tools to simplify Stokes' Theorem. But geometric objects like forms should be visualized! Here, I present visualizations of forms that attempt to capture their geometry. Contents Introduction 1 1. Exterior algebra 2 1.1. Vectors and dual vectors 2 1.2. The wedge of two vectors 3 1.3. Defining the exterior algebra 4 1.4. The wedge of two dual vectors 5 1.5. R3 has no mixed wedges 6 1.6. Interior multiplication 7 2. Integration 8 2.1. Differential forms 9 2.2. Visualizing differential forms 9 2.3. Integrating differential forms over manifolds 9 3. Differentiation 11 3.1. Exterior Derivative 11 3.2. Visualizing the exterior derivative 13 3.3. Closed and exact forms 14 3.4. Future work: Lie derivative 15 Acknowledgments 16 References 16 Introduction What's the natural way to integrate over a smooth manifold? Well, a k-dimensional manifold is (locally) parametrized by k coordinate functions. At each point we get k tangent vectors, which together form an infinitesimal k-dimensional volume ele- ment. And how should we measure this element? It seems reasonable to ask for a function f taking k vectors, so that • f is linear in each vector • f is zero for degenerate elements (i.e. when vectors are linearly dependent) • f is smooth These are the properties of a differential form. 1 2 GREGORY BIXLER Introducing structure to a manifold usually follows a certain pattern: • Create it in Rn. We define the exterior algebra in section 1. • Introduce it locally in coordinate neighborhoods. We devote sections 2 and 3 to this. • Understand it globally by patching together the coordinate neighborhoods. This is beyond the scope of this paper, except for a few remarks at the end of section 3. 1. Exterior algebra Given a vector space1 V , we can construct its exterior algebra2 Λ(V ), a se- quence of vector spaces Λk(V ) which represent k-dimensional volume elements.We measure these volume elements via the dual space Λk(V )_. This generalizes the line integrals seen in a typical introductory multivariable calculus class: R F · dr. Here, \F ·" is a dual vector measuring the line element dr. We face a double challenge: to see the exterior algebra of dual vectors. We'll tackle this by first visualizing dual vectors, and then visualizing the exterior algebra of (regular) vectors and dual vectors in tandem. 1.1. Vectors and dual vectors. Visually, a vector in Rn is an arrow whose tail is at the origin. It's less clear how to visualize dual vectors. Since there is an isomorphism V _ =∼ V , why not simply visualize V _ as V itself? The main problem with this approach is that it transforms the wrong way under a change of basis. For example, suppose we're initially measuring in feet and switch to inches: if our old basis is fe1; : : : ; eng, then our new basis is fe1=12; : : : ; en=12g. Vectors appear to get 12 times bigger. But for any T 2 V _ and v 2 V , the value of T (v) is independent of basis, so T must shrink by a factor of 12, not grow. Instead, in R1, view a dual vector a collection of \tick marks" on the axis: In Rn, we can do the same, with a caveat. We must choose an axis with tick marks, and project the vector into this axis. 1 In this paper, all vector spaces are finite-dimensional. 2 The notation “Λ” seems to be a corruption of the symbol ^, the exterior/wedge product. Some authors write V(V ) instead. VISUALIZING EXTERIOR CALCULUS 3 This doesn't really measure the vector: it measures the projection. We can measure vectors more directly by orthogonally extending the tick marks. This yields a series of lines in R2, planes in R3, and hyperplanes in general. Formally, we can think of this geometric representation of a dual vector T as the set of pre-images T −1(n) of integers n. 1.2. The wedge of two vectors. Allow me to start with a historical note. In R3, in addition to the vector space operations of addition and scaling, we have the dot product and cross product. We compute the dot product u · v by projecting u along v and then multiplying their signed lengths. We compute the cross product u × v by first forming the parallelogram P spanned by u and v and then setting the length of u × v to be the area of P , in the direction of the (oriented) normal to P . These operations were not always called \dot product" and \cross product". Hermann Grassman essentially invented linear algebra in his 1844 book Die lineale Ausdehnungslehre ([3]), including the exterior algebra. In the preface, he calls these the \interior product"3 and \exterior product," respectively. He reasons that the dot product takes its maximum value when one vector is inside the span of the other, while the cross product is only nonzero if each vector is outside the span of the other. 3 This is not the same as the interior multiplication we define later! Also, this is probably where the term \inner product" comes from. 4 GREGORY BIXLER Anyways, back to describing multi-dimensional volume. The cross product is essentially the right idea|it's no accident that it shows up whenever we compute a surface integral. To generalize it, we need to fix a quirk of the cross product: it describes a 2-dimensional area as a 1-dimensional vector. The fix is to describe the result as a \2-dimensional vector"!4 We implement this fix by defining a new operation, the wedge product (or exterior product) u ^ v of vectors u and v. We want it to satisfy many of the same properties as the cross product: • (u1 + u2) ^ v = u1 ^ v + u2 ^ v • c(u ^ v) = (cu) ^ v = u ^ (cv) • u ^ v = −(v ^ u), so in particular, v ^ v = 0 1.3. Defining the exterior algebra. The wedge product looks essentially like the tensor product, but with some extra relations (in the third bullet above). So, it should seem natural to define the space of 2-vectors, Λ2(V ), as a quotient of the tensor power V ⊗ V . Before we do this, recall a little trick, which shows that the relations v ^ v = 0 imply u ^ v = −v ^ u. 0 = (u + v) ^ (u + v) = u ^ u + u ^ v + v ^ u + v ^ v = u ^ v + v ^ u: We now define V ⊗ V Λ2(V ) = ; I where I is the subspace of V ⊗ V generated by the tensors v ⊗ v. We can generalize this definition to higher powers: Definition 1.1. Let V be a vector space, and k a nonnegative integer. Then V ⊗k Λk(V ) = ; Ik ⊗k where Ik is the subspace of V generated by tensors v1 ⊗ · · · ⊗ vk, where at least one of the vi is repeated. (Or equivalently, where the vi are linearly dependent.) Recall that, if fe1; : : : ; eng is a basis for V , then the elements eI = ei1 ^ · · · ^ eik form a basis for Λk(V ). Here, I ranges over all strictly increasing k-tuples in k n f1; : : : ; ng. The dimension of Λ (V ) is the number of such tuples, which is k . Finally, treating the exterior algebra as a whole makes the definition even simpler: Let T (V ) be the tensor algebra for V . Then Λ(V ) = T (V )=I; where I is the ideal generated by the tensors v ⊗ v.5 4 3 Perhaps I'm being unfair to the cross product here. It really is remarkable that in R , we can describe area using vectors, and the cross product is particularly useful since it spits out an object of the same type as its inputs. 5 The ideal I is different from the subspace I from the definition of Λ2(V ), since now we also use ⊗ when generating I. VISUALIZING EXTERIOR CALCULUS 5 1.4. The wedge of two dual vectors. We've seen that the wedge of two vectors looks like a stretchy, oriented parallelogram. How about the wedge of two dual vectors? First, recall the usual definition of 2-forms. Proposition 1.2. Let V be a finite-dimensional vector space. Then Λ2(V _) = W; 6 where W is the space of bilinear, alternating functions V × V ! R. Let's start simply, in R2. Here, we have a way to measure oriented parallelo- grams: the determinant. Indeed, the function (u; v) 7! det u v is bilinear and alternating. The determinant in R2 is a particular 2-form; recall n _ _ _ that in general, the determinant detn in R is equal to e1 ^ · · · ^ en , where ej is the usual basis for (Rn)_. Here's how we can visualize the determinant: Place a dot at each lattice point in the plane. To approximate det(u ^ v), draw the parallelogram u ^ v and count how many dots are inside. Keep track of orientations with a small circle around each dot. For example, in the below figure, det(u1 ^ u2) ≈ 3 and det(v1 ^ v2) ≈ −4. From now on, I'll mostly ignore orientations. To rescale the determinant, move the dots closer together or farther apart, reversing the circles if the scalar is negative. Due to the bilinearity of the determinant, it doesn't matter how you choose to move the dots. 6 A function is alternating if its sign flips when swapping any two of its arguments.